In order to keep the answer as accurate as possible, I will not refer to the staircase in the English form, but will refer to it as "la escalera", to keep the significance that Buero Vallejo intends for it to have.
In Antonio Buero Vellejo's Historia de una Escalera, "la escalera", is the one consistent motif that appears at all times near, in front, behind or underneath the characters.
"La escalera" is a character, in itself. As an object through which people go up and down to commute, "la escalera" stands for the constant need for mobility in everyday life. However, as it remains the same and it does not go anywhere in the play, its meaning becomes juxtaposed, representing then stagnation, and the inability to move forward (or away).
"La escalera" also means the passing of time. As an object, it helps us go from one location to another. Hence, it symbolically represents how the tenement has witnessed the passing of time as well, from one generation to another. However, even though the time span of the play is thirty years, we find that the different inhabitants of the tenement continue to make the same social mistakes of the previous generations: people continue to want for the same things, to complaint about the same matters and society, as a whole, does not get any better.
All of these different people of all types of backgrounds go up and down the same escalera. Therefore, whether they go up, or down, their social failures remain. That social failure is another symbolism that can be awarded to it.
Historically speaking, Buero Vallejo writes this play after the havoc that is caused by the Spanish Civil war. This war disenfranchised its own people by depleting them of their self-pride, and by squandering the economy. As a result, the atmosphere of the play is one of desolation, angst, depression, and frustration in the horizon.
Therefore, the significance of the title Historia de una Escalera is that this one object is personified as a witness of time, society, and history. It is also the one firm thing that remains after the social inequities that take place over the thirty years that the play represents.The escalera has seen it all, it has carried them all up and down their daily lives, and still remains to see what will happen next.